1. Field of the Invention
The present invention relates to control for aerial vehicles, and particularly to an adaptive control method for aerial vehicles carrying slung, or suspended, loads. (u, v, w)
2. Description of the Related Art
FIGS. 4 and 5 illustrate an exemplary aerial vehicle 102. In relation to discussing the aerial vehicle 102 with reference to FIGS. 4 and 5, the symbols, letters, characters and references used are for purposes of the background description of the related art only, and the meaning or use of such same or similar symbols, letters, characters and references may differ as otherwise described in the specification. The aerial vehicle 102 is modeled as a rigid body with six degrees of freedom. The twelve aerial vehicle states include translational velocities (u, v, w) angular velocities (p, q, r), Euler angles (Φ, θ, φ) and the aerial vehicle 102's position (x, y, z). An external load is modeled as a point mass that behaves like a spherical pendulum suspended from a single point. The cable for the external load is assumed to be inelastic and with no mass. The geometry and the relevant coordinate systems are shown in FIG. 4. The unit vectors iH, jH, kH of the “hook” coordinate system always remain parallel to those of the body axis system of aerial vehicle 102. The position of the load is described by the two angles ΦL and θL, where θL is load angle in the x-z plane, and where ΦL is the load oscillation angle out of the x-z plane. Therefore, the position vector RL of the load with respect to the suspension point is given by:RL=L cos(θL)sin(ΦL)iH+L sin(θL)jH+L cos(θL)cos(ΦL)kH.  (1)
The position vector RH of the hook with respect to the aerial vehicle 102's center of gravity (e.g.) is given by:RH=xHiH+yHjH+zHkH.  (2)The absolute velocity VL of the load is given by:VL=Vcg+{dot over (R)}+Ω×R,  (3)where Vcg is the absolute velocity of the center of mass of the aerial vehicle 102, R=RL+RH is the position vector of the load with respect to the center of mass of the aerial vehicle 102, and Q=piH+qjH+rkH is the angular velocity of the aerial vehicle 102. The absolute acceleration aL of the load is:aL={dot over (V)}L+Ω×VL.  (4)The unit vector in the direction of the gravity force is given by:Kg=−sin(θ)iH+sin(Φ)cos(θ)jH+cos(φ)cos(θ)kH.  (5)Beside the gravity, there is an aerodynamic force applied on the point mass load. Since the analysis in the background description is restricted to the aerial vehicle 102's motion near hover, the aerodynamics loads on the load will be neglected.
The equations of motion of the load are written by enforcing moment equilibrium about the suspension point, that is, in matrix form:RL×(−mLaL+mLgkg)=0.  (6)The above equation gives three scalar equations of second order, only the equations in the x and y directions are retained, which represent the equations of motion of the load.
The suspended load introduces additional terms in the rigid body force and moment equations of motion of the aerial vehicle 102, namely load forces and load dynamics. The force and moment loads, FHL and MHL, are shown in FIG. 5. The force that the load exerts on the aerial vehicle 102 is given by:FHL=−mLaL+mLgkg.  (7)The additional moment MHL is therefore given by:MHL=RH×FHL.  (8)The above equations give highly nonlinear expressions. These equations cannot be used for stability analysis. Therefore, they must be linearized around the trim condition. To be able to perform the linearization process, the trim values of the aerial vehicle 102 and the load must be determined.
The obtained equations are nonlinear and complicated. For design purposes, these equations are linearized about the hovering conditions. Near hover, the forward speed is nearly zero (i.e., u0=0). Assuming that the aerial vehicle 102's roll angle is also zero, even with the effect of the load on the aerial vehicle 102 (i.e., Φ0=0) simplifies the analysis. At this condition, the load trim equations give the following trim values:θLo0 and ΦLo=−θo.  (9)Imposing the above results to the linearized load equations obtains the following equations of motion for the load:gθL−g cos(θo)φ+yh{dot over (q)}+{dot over (V)}+L{umlaut over (θ)}L=0.  (10)L{umlaut over (Φ)}L+gΦL+gθ+(xh−L sin(θo))cos(θo){dot over (p)}+zh sin(θo){dot over (r)}+L cos(θo)sin(θo){dot over (r)}+cos (θo){dot over (U)}+sin(θo){dot over (W)}=0.  (11)The forces exerted by the load on the aerial vehicle 102 are:Fx=mL(−g cos[θO]θ−xh{dot over (p)}+L sin[θo]{dot over (p)}−{dot over (U)}[t]−L cos[θo]{umlaut over (Φ)}L),Fy=mL(g cos[θo]Φ−yh{dot over (q)}−{dot over (V)}−L{umlaut over (θ)}L) andFz=mL(−g sin[θo]θ−(zh+L cos[θo]){dot over (r)}−{dot over (W)}−L sin[θo]{umlaut over (Φ)}L).  (12)The moments in the x-y-z directions are: φ
                                          M            x                    =                                    m              L                        ⁡                          (                                                                                                                                            -                          g                                                ⁢                                                                                                  ⁢                                                  y                          h                                                ⁢                                                  sin                          ⁡                                                      [                                                          θ                              o                                                        ]                                                                          ⁢                        θ                                            -                                              g                        ⁢                                                                                                  ⁢                                                  z                          h                                                ⁢                                                  cos                          ⁡                                                      (                                                          θ                              o                                                        )                                                                          ⁢                        φ                                            +                                                                        y                          h                                                ⁢                                                  z                          h                                                ⁢                                                  q                          .                                                                                                                                                                                -                                                                        y                          h                                                (                                                                              z                            h                                                    +                                                                                    Ly                              h                                                        ⁢                                                          cos                              ⁡                                                              [                                                                  θ                                  o                                                                ]                                                                                      ⁢                                                          r                              .                                                                                +                                                                                    z                              h                                                        ⁢                                                          V                              .                                                                                -                                                                                    y                              h                                                        ⁢                                                          W                              .                                                                                                                                                                                                                                                                                        +                          L                                                ⁢                                                                                                  ⁢                                                  z                          h                                                ⁢                                                  θ                          ¨                                                                    -                                              L                        ⁢                                                                                                  ⁢                                                  y                          h                                                ⁢                                                  sin                          ⁡                                                      [                                                          θ                              o                                                        ]                                                                          ⁢                                                                              φ                            ¨                                                    L                                                                                                                                )                                      ⁢                                  ⁢                              M            y                    =                                    m              L                        ⁡                          (                                                                                                                                            -                          g                                                ⁢                                                                                                  ⁢                                                  z                          h                                                ⁢                                                  cos                          ⁡                                                      [                                                          θ                              o                                                        ]                                                                          ⁢                        θ                                            -                                              g                        ⁢                                                                                                  ⁢                                                  x                          h                                                ⁢                                                  sin                          ⁡                                                      [                                                          θ                              o                                                        ]                                                                          ⁢                                                  m                          L                                                ⁢                        θ                                            -                                                                        x                          h                                                ⁢                                                  z                          h                                                ⁢                                                  m                          L                                                ⁢                                                  p                          .                                                                                                                                                                                                                                                              +                            L                                                    ⁢                                                                                                          ⁢                                                      z                            h                                                    ⁢                                                      sin                            ⁡                                                          [                                                              θ                                o                                                            ]                                                                                ⁢                                                      p                            .                                                                          +                                                                              x                            h                                                    ⁢                                                      z                            h                                                    ⁢                                                      r                            .                                                                          +                                                  L                          ⁢                                                                                                          ⁢                                                      x                            h                                                    ⁢                                                      cos                            ⁡                                                          [                                                              θ                                o                                                            ]                                                                                ⁢                                                      r                            .                                                                          -                                                                              z                            h                                                    ⁢                                                      U                            .                                                                                              ⁢                                                                                                                                                                                                                                                        +                                                      x                            h                                                                          ⁢                        W                                            -                                              L                        ⁢                                                                                                  ⁢                                                  z                          h                                                ⁢                                                  cos                          ⁡                                                      [                                                          θ                              o                                                        ]                                                                          ⁢                                                                              φ                            ¨                                                    L                                                                    +                                              L                        ⁢                                                                                                  ⁢                                                  x                          h                                                ⁢                                                  sin                          ⁡                                                      [                                                          θ                              o                                                        ]                                                                          ⁢                                                                              φ                            ¨                                                    L                                                                                                                                )                                      ⁢                                  ⁢        and        ⁢                                  ⁢                              M            z                    =                                                    m                L                            ⁡                              (                                                                                                                              g                          ⁢                                                                                                          ⁢                                                      y                            h                                                    ⁢                                                      cos                            ⁡                                                          [                                                              θ                                o                                                            ]                                                                                ⁢                          θ                                                +                                                  g                          ⁢                                                                                                          ⁢                                                      x                            h                                                    ⁢                                                      cos                            ⁡                                                          (                                                              θ                                o                                                            )                                                                                ⁢                          φ                                                +                                                                              x                            h                                                    ⁢                                                      y                            h                                                    ⁢                                                      p                            .                                                                                                                                                                                                                                                        -                            L                                                    ⁢                                                                                                          ⁢                                                      y                            h                                                    ⁢                                                      sin                            ⁡                                                          [                                                              θ                                o                                                            ]                                                                                ⁢                                                      p                            .                                                                          -                                                                              x                            h                                                    ⁢                                                      y                            h                                                    ⁢                                                      q                            .                                                                          +                                                                              y                            h                                                    ⁢                                                      U                            .                                                                          -                                                                              x                            h                                                    ⁢                                                      V                            .                                                                                                                                                                                                                                                        -                            L                                                    ⁢                                                                                                          ⁢                                                      x                            h                                                    ⁢                                                                                    θ                              ¨                                                        L                                                                          +                                                  L                          ⁢                                                                                                          ⁢                                                      y                            h                                                    ⁢                                                      cos                            ⁡                                                          [                                                              θ                                o                                                            ]                                                                                ⁢                                                                                    φ                              ¨                                                        L                                                                                                                                              )                                      .                                              (        13        )            
These equations are linear and can be formulated in a state space form. If the load state vector is defined as xL=[{dot over (Φ)}L {dot over (θ)}L ΦL θL ]T, the load equations in state space can be written as:EL{dot over (x)}=ALx,  (14)where x is the state vector for the load and the aerial vehicle 102 (i.e., x=[xH xL]).Similarly, the effect of the load on the aerial vehicle 102 force terms can be written also as:
                              [                                                                      F                  HL                                                                                                      M                  HL                                                              ]                =                                            E              HL                        ⁢                          x              .                                +                                    A              HL                        ⁢                          x              .                                                          (        15        )            
The linearized equations of motion of the aerial vehicle 102 and the load can be written in the following state space forms:
                              x          .                =                              [                                                                                                      x                      .                                        h                                                                                                                                          x                      .                                        L                                                                        ]                    =                                    A              ⁢                                                          ⁢              x                        +                          B              ⁢                                                          ⁢                              η                .                                                                        (        16        )            A great deal of effort has been made for modeling the slung load and studying its effect on dynamics of aerial vehicles. Examples of automatic control for vehicles, such as helicopters, with slung loads include a single-cable dynamic model developed using a straightforward application of Lagrange equations, and an expanded version of this model, which includes two tandem cables. However, such a formulation was based on the Newton-Euler equations of motion for small perturbations separated into longitudinal and lateral sets. The disadvantage was that aerodynamic forces on the cables and the load were neglected, as were the helicopter rotor dynamic modes.
Such work, though promising, is not only based on classical control techniques, but is difficult to apply to modem unmanned aerial vehicles, such as quadrotor drones and the like. It would be desirable to provide an anti-swing controller for a quadrotor aerial vehicle slung load system near hover flight. Such a controller should be based on time-delayed feedback of the load swing angles. The output from such a controller would be additional displacements that are added to the vehicle trajectory in the longitudinal and lateral directions, for example.
Thus, an adaptive control method for an unmanned vehicle with a slung load addressing the aforementioned problems is desired.